The Laplace transform of the Caputo's fractional differentiation operator [1]} is defined by
\( {L} \left\lbrace ({C}{}D^{\alpha }_{0^{+}}g)(t)\right\rbrace (s) =s^{\alpha }G(s)-\sum _{k=1}^{n}s^{\alpha -k}g^{(k-1)}(0), \quad n-1<\alpha \le n, \quad n\in \mathbb {N},\) 
